Learn · updated 2026-10-23

Control chart constants: A2, D3, D4, d2, c4 and the rest

By Nicholas Williford, Lean Six Sigma Black Belt · About LimitLine

Control limits sit three standard deviations of the plotted statistic from the center line, but on an X̄-R or X̄-S chart you estimate the standard deviation from ranges or subgroup standard deviations. The constants turn those averages into limits, so you only multiply.

The table

Subgroup sizes 2 to 25. Use the row for the number of parts in each subgroup.

For X̄-R charts (ranges)

For X̄-S charts (standard deviations)

A2, D3 and D4 for n = 2 to 10 agree with NIST's printed table. The one difference is D4 for n = 5: the exact value is 2.1145, which NIST prints as 2.115 and this table rounds to 2.114; either gives the same limits in practice. Every value is recalculated from the definitions below each time this page is built.

Worked example

Say you measure 5 parts an hour and after 25 hours have X̿ = 10.00 mm and R̄ = 0.20 mm. From the n = 5 row:

1. X̄ chart: 10.00 ± 0.577 × 0.20 → UCL 10.115, LCL 9.885.
2. R chart: UCL = 2.114 × 0.20 = 0.423; LCL = 0 (D3 is 0 for n ≤ 6).
3. Process σ = 0.20 / 2.326 = 0.0860 mm, the σ to use for Cp and Cpk.

The X̄-R chart guide works a full example with real subgroups.

Where they come from

• d2 is the average range of n values from a normal distribution with σ = 1, so σ ≈ R̄ / d2. d3 is the standard deviation of that range.
• c4 corrects the subgroup standard deviation, which runs a little low on average: σ ≈ s̄ / c4. It is √(2/(n−1)) × Γ(n/2) / Γ((n−1)/2).
• A2 = 3 / (d2 √n): three standard errors of a subgroup mean, in units of R̄. A3 = 3 / (c4 √n), the same in units of s̄.
• D3, D4 = 1 ∓ 3 d3 / d2, and B3, B4 = 1 ∓ 3 √(1 − c4²) / c4. A negative lower value is set to 0, which is why small subgroups have a lower limit of 0 on the R or S chart.
• 2.66 on the I-MR chart is 3 / d2 for n = 2 (3 / 1.128), and 3.267 is D4 for n = 2.

R or S?

The range throws away the values between the largest and smallest, which costs little with small subgroups and more as they grow. NIST puts the range's efficiency at 0.955 for n = 5 and 0.850 for n = 10. Use X̄-R up to about 9 parts per subgroup and X̄-S from 10.

Sources

1. NIST/SEMATECH e-Handbook of Statistical Methods, §6.3.2.1 Shewhart X-bar and R and S Control Charts (A2, D3, D4 table; efficiency of R)